In a RNA-seq dataset, I have two batches that have one sample in common, plus other samples in each batch. There are 3 replicates for each sample. Is there a way to compare other samples across tthe two batches? In other words, how can the batch effects be removed? The setup is shown above, how can I do DEG analysis to compare sample1 to sample3, or sample 2 to sample4?

The short answer is yes, you can use even a single sample to estimate the batch effect (and therefore control for it in otherwise confounded comparison).

However you have low power with a single sample, as compared to an orthogonal (balanced) design. This is because there is substantial variance from using the single sample to define the batch effect, and therefore also the condition effect.